Circle Theorems

Learning Objective:

• we are learning to recall, understand, apply, prove and extend the eight standard circle theorems

Investing:

• This is useful because we need to be able to find angles in and around circles
• A functional (real-life) application is in engineering and design (cogs and cams?); navigation?; astronomy? [citations needed]
• This skill leads to developing our skills at geometric proof
• A 'free gift' with this skill is that we can now also ...
• This could help you if you want to work in engineering, design, ...

Preparing:

• Can you already find angles in arrangements of straight lines including those that use the properties of systems of parallel lines and angles in triangles including the isosceles property?
• Let's be sure you can by [create flashcard deck?]...
• Before you start you need to know key vocabulary for circles and parts of circles:
  • circumference
  • diameter
  • radius
  • chord
  • tangent
  • segment
  • sector
  • arc
  • major and minor as adjectives before segment, sector and arc
    
    
• You will have a deeper understanding if you also know the proofs for various angle rules including:
  • given that the angles at a point (Y shape) sum to 360°, the angles at a point on a straight line (K shape) sum to 180°
  • given the angles on a straight line sum to 180°, then 'vertically opposite' angles (X shape) are equal
  • given that corresponding angles when a transversal crosses a pair of parallel lines (F shape) are equal and the (Y, K and X) properties above, that alternate angles (Z shape) are also equal
  • given that corresponding angles when a transversal crosses a pair of parallel lines (F shape) are equal and the (Y, K and X) properties above, that allied angles (a.k.a. 'consecutive interior angles' making a [ or C shape) are also equal
  • given that alternate angles when a transversal crosses a pair of parallel lines (Z shape) are equal and the (Y, K and X) properties above, that interior angles of a triangle sum to 180°.

Discovering Theorem 1:

• Can you figure it out yourself from these examples?
• Focus on the red chord and the angle it makes with the red radius:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 1 - a radius perpendicular to a chord bisects the chord AND a chord which is bisected by a radius is perpendicular to that radius.

Discovering Theorem 2:

• Can you figure it out yourself from these examples?
• Focus on the red chord and the angle it makes with the red radius:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 2 - a tangent is always perpendicular to the radius which meets it.

Discovering Theorem 3:

• Can you figure it out yourself from these examples?

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 3 - two tangents which meet at a point are equal in length.

Discovering Theorem 4:

• Can you figure it out yourself from these examples?
• Only move the slider for point A at the moment:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 4 - the angle in a semicircle is always 90°.

Discovering Theorem 5:

• Can you figure it out yourself from these examples?
• Now you can generalise Theorem 2 by moving the other points:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 5 - the angle at the centre is always double the angle at the circumference.

Discovering Theorem 6:

• Can you figure it out yourself from these examples?
• Now you can generalise Theorem 2 in a different way by moving the other points C and D:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 6 - all the angles drawn from a chord in the same segment are equal.

Discovering Theorem 7:

• Can you figure it out yourself from these examples?
• Now you can generalise Theorem 2 by moving the other points:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 7 - opposite angles in a cyclic quadrilateral sum to 180°.

Discovering Theorem 8:

• Can you figure it out yourself from these examples?
• Now you can generalise Theorem 2 by moving the other points:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 8 - ... and the other pair of opposite angles in a cyclic quadrilateral sum to 180° too.

Discovering Theorem 9:

• Can you figure it out yourself from these examples?

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Theorem 9 - the 'alternate segment theorem' - the angle between a chord and a tangent that meets one of the chord's endpoints equals the angle in the alternate segment.

Summary:

Modeling:

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:
• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

• Three presentations to help you memorize the circle theorems

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...

last edited: Oct 9, 2015 4:03 am